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The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if ${U: H \rightarrow H}$ is a unitary operator on a Hilbert space ${H}$, and ${v \in H}$ is a vector in that Hilbert space, then one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N U^n v = \pi_{H^U} v$

in the strong topology, where ${H^U := \{ w \in H: Uw = w \}}$ is the ${U}$-invariant subspace of ${H}$, and ${\pi_{H^U}}$ is the orthogonal projection to ${H^U}$. (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if ${G}$ is a countable amenable group acting on a Hilbert space ${H}$ by unitary transformations ${g: H \rightarrow H}$, and ${v \in H}$ is a vector in that Hilbert space, then one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv = \pi_{H^G} v \ \ \ \ \ (1)$

for any Folner sequence ${\Phi_N}$ of ${G}$, where ${H^G := \{ w \in H: gw = w \hbox{ for all }g \in G \}}$ is the ${G}$-invariant subspace. Thus one can interpret ${\pi_{H^G} v}$ as a certain average of elements of the orbit ${Gv := \{ gv: g \in G \}}$ of ${v}$.

I recently discovered that there is a simple variant of this ergodic theorem that holds even when the group ${G}$ is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem) Let ${G}$ be an arbitrary group acting unitarily on a Hilbert space ${H}$, and let ${v}$ be a vector in ${H}$. Then ${\pi_{H^G} v}$ is the element in the closed convex hull of ${Gv := \{ gv: g \in G \}}$ of minimal norm, and is also the unique element of ${H^G}$ in this closed convex hull.

Proof: As the closed convex hull of ${Gv}$ is closed, convex, and non-empty in a Hilbert space, it is a classical fact (see e.g. Proposition 1 of this previous post) that it has a unique element ${F}$ of minimal norm. If ${T_g F \neq F}$ for some ${g}$, then the midpoint of ${T_g F}$ and ${F}$ would be in the closed convex hull and be of smaller norm, a contradiction; thus ${F}$ is ${G}$-invariant. To finish the first claim, it suffices to show that ${v-F}$ is orthogonal to every element ${h}$ of ${H^G}$. But if this were not the case for some such ${h}$, we would have ${\langle T_g v - F, h \rangle = \langle v-F,h\rangle \neq 0}$ for all ${g \in G}$, and thus on taking convex hulls ${\langle F-F,h\rangle = \langle f-F,f\rangle \neq 0}$, a contradiction.

Finally, since ${T_g v - F}$ is orthogonal to ${H^G}$, the same is true for ${F'-F}$ for any ${F'}$ in the closed convex hull of ${Gv}$, and this gives the second claim. $\Box$

This result is due to Alaoglu and Birkhoff. It implies the amenable ergodic theorem (1); indeed, given any ${\epsilon>0}$, Theorem 1 implies that there is a finite convex combination ${v_\epsilon}$ of shifts ${gv}$ of ${v}$ which lies within ${\epsilon}$ (in the ${H}$ norm) to ${\pi_{H^G} v}$. By the triangle inequality, all the averages ${\frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv_\epsilon}$ also lie within ${\epsilon}$ of ${\pi_{H^G} v}$, but by the Folner property this implies that the averages ${\frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv}$ are eventually within ${2\epsilon}$ (say) of ${\pi_{H^G} v}$, giving the claim.

It turns out to be possible to use Theorem 1 as a substitute for the mean ergodic theorem in a number of contexts, thus removing the need for an amenability hypothesis. Here is a basic application:

Corollary 2 (Relative orthogonality) Let ${G}$ be a group acting unitarily on a Hilbert space ${H}$, and let ${V}$ be a ${G}$-invariant subspace of ${H}$. Then ${V}$ and ${H^G}$ are relatively orthogonal over their common subspace ${V^G}$, that is to say the restrictions of ${V}$ and ${H^G}$ to the orthogonal complement of ${V^G}$ are orthogonal to each other.

Proof: By Theorem 1, we have ${\pi_{H^G} v = \pi_{V^G} v}$ for all ${v \in V}$, and the claim follows. (Thanks to Gergely Harcos for this short argument.) $\Box$

Now we give a more advanced application of Theorem 1, to establish some “Mackey theory” over arbitrary groups ${G}$. Define a ${G}$-system ${(X, {\mathcal X}, \mu, (T_g)_{g \in G})}$ to be a probability space ${X = (X, {\mathcal X}, \mu)}$ together with a measure-preserving action ${(T_g)_{g \in G}}$ of ${G}$ on ${X}$; this gives an action of ${G}$ on ${L^2(X) = L^2(X,{\mathcal X},\mu)}$, which by abuse of notation we also call ${T_g}$:

$\displaystyle T_g f := f \circ T_{g^{-1}}.$

(In this post we follow the usual convention of defining the ${L^p}$ spaces by quotienting out by almost everywhere equivalence.) We say that a ${G}$-system is ergodic if ${L^2(X)^G}$ consists only of the constants.

(A technical point: the theory becomes slightly cleaner if we interpret our measure spaces abstractly (or “pointlessly“), removing the underlying space ${X}$ and quotienting ${{\mathcal X}}$ by the ${\sigma}$-ideal of null sets, and considering maps such as ${T_g}$ only on this quotient ${\sigma}$-algebra (or on the associated von Neumann algebra ${L^\infty(X)}$ or Hilbert space ${L^2(X)}$). However, we will stick with the more traditional setting of classical probability spaces here to keep the notation familiar, but with the understanding that many of the statements below should be understood modulo null sets.)

A factor ${Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})}$ of a ${G}$-system ${X = (X,{\mathcal X},\mu, (T_g)_{g \in G})}$ is another ${G}$-system together with a factor map ${\pi: X \rightarrow Y}$ which commutes with the ${G}$-action (thus ${T_g \pi = \pi S_g}$ for all ${g \in G}$) and respects the measure in the sense that ${\mu(\pi^{-1}(E)) = \nu(E)}$ for all ${E \in {\mathcal Y}}$. For instance, the ${G}$-invariant factor ${Z^0_G(X) := (X, {\mathcal X}^G, \mu\downharpoonright_{{\mathcal X}^G}, (T_g)_{g \in G})}$, formed by restricting ${X}$ to the invariant algebra ${{\mathcal X}^G := \{ E \in {\mathcal X}: T_g E = E \hbox{ a.e. for all } g \in G \}}$, is a factor of ${X}$. (This factor is the first factor in an important hierachy, the next element of which is the Kronecker factor ${Z^1_G(X)}$, but we will not discuss higher elements of this hierarchy further here.) If ${Y}$ is a factor of ${X}$, we refer to ${X}$ as an extension of ${Y}$.

From Corollary 2 we have

Corollary 3 (Relative independence) Let ${X}$ be a ${G}$-system for a group ${G}$, and let ${Y}$ be a factor of ${X}$. Then ${Y}$ and ${Z^0_G(X)}$ are relatively independent over their common factor ${Z^0_G(Y)}$, in the sense that the spaces ${L^2(Y)}$ and ${L^2(Z^0_G(X))}$ are relatively orthogonal over ${L^2(Z^0_G(Y))}$ when all these spaces are embedded into ${L^2(X)}$.

This has a simple consequence regarding the product ${X \times Y = (X \times Y, {\mathcal X} \times {\mathcal Y}, \mu \times \nu, (T_g \oplus S_g)_{g \in G})}$ of two ${G}$-systems ${X = (X, {\mathcal X}, \mu, (T_g)_{g \in G})}$ and ${Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})}$, in the case when the ${Y}$ action is trivial:

Lemma 4 If ${X,Y}$ are two ${G}$-systems, with the action of ${G}$ on ${Y}$ trivial, then ${Z^0_G(X \times Y)}$ is isomorphic to ${Z^0_G(X) \times Y}$ in the obvious fashion.

This lemma is immediate for countable ${G}$, since for a ${G}$-invariant function ${f}$, one can ensure that ${T_g f = f}$ holds simultaneously for all ${g \in G}$ outside of a null set, but is a little trickier for uncountable ${G}$.

Proof: It is clear that ${Z^0_G(X) \times Y}$ is a factor of ${Z^0_G(X \times Y)}$. To obtain the reverse inclusion, suppose that it fails, thus there is a non-zero ${f \in L^2(Z^0_G(X \times Y))}$ which is orthogonal to ${L^2(Z^0_G(X) \times Y)}$. In particular, we have ${fg}$ orthogonal to ${L^2(Z^0_G(X))}$ for any ${g \in L^\infty(Y)}$. Since ${fg}$ lies in ${L^2(Z^0_G(X \times Y))}$, we conclude from Corollary 3 (viewing ${X}$ as a factor of ${X \times Y}$) that ${fg}$ is also orthogonal to ${L^2(X)}$. Since ${g}$ is an arbitrary element of ${L^\infty(Y)}$, we conclude that ${f}$ is orthogonal to ${L^2(X \times Y)}$ and in particular is orthogonal to itself, a contradiction. (Thanks to Gergely Harcos for this argument.) $\Box$

Now we discuss the notion of a group extension.

Definition 5 (Group extension) Let ${G}$ be an arbitrary group, let ${Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})}$ be a ${G}$-system, and let ${K}$ be a compact metrisable group. A ${K}$-extension of ${Y}$ is an extension ${X = (X, {\mathcal X}, \mu, (T_g)_{g \in G})}$ whose underlying space is ${X = Y \times K}$ (with ${{\mathcal X}}$ the product of ${{\mathcal Y}}$ and the Borel ${\sigma}$-algebra on ${K}$), the factor map is ${\pi: (y,k) \mapsto y}$, and the shift maps ${T_g}$ are given by

$\displaystyle T_g ( y, k ) = (S_g y, \rho_g(y) k )$

where for each ${g \in G}$, ${\rho_g: Y \rightarrow K}$ is a measurable map (known as the cocycle associated to the ${K}$-extension ${X}$).

An important special case of a ${K}$-extension arises when the measure ${\mu}$ is the product of ${\nu}$ with the Haar measure ${dk}$ on ${K}$. In this case, ${X}$ also has a ${K}$-action ${k': (y,k) \mapsto (y,k(k')^{-1})}$ that commutes with the ${G}$-action, making ${X}$ a ${G \times K}$-system. More generally, ${\mu}$ could be the product of ${\nu}$ with the Haar measure ${dh}$ of some closed subgroup ${H}$ of ${K}$, with ${\rho_g}$ taking values in ${H}$; then ${X}$ is now a ${G \times H}$ system. In this latter case we will call ${X}$ ${H}$-uniform.

If ${X}$ is a ${K}$-extension of ${Y}$ and ${U: Y \rightarrow K}$ is a measurable map, we can define the gauge transform ${X_U}$ of ${X}$ to be the ${K}$-extension of ${Y}$ whose measure ${\mu_U}$ is the pushforward of ${\mu}$ under the map ${(y,k) \mapsto (y, U(y) k)}$, and whose cocycles ${\rho_{g,U}: Y \rightarrow K}$ are given by the formula

$\displaystyle \rho_{g,U}(y) := U(gy) \rho_g(y) U(y)^{-1}.$

It is easy to see that ${X_U}$ is a ${K}$-extension that is isomorphic to ${X}$ as a ${K}$-extension of ${Y}$; we will refer to ${X_U}$ and ${X}$ as equivalent systems, and ${\rho_{g,U}}$ as cohomologous to ${\rho_g}$. We then have the following fundamental result of Mackey and of Zimmer:

Theorem 6 (Mackey-Zimmer theorem) Let ${G}$ be an arbitrary group, let ${Y}$ be an ergodic ${G}$-system, and let ${K}$ be a compact metrisable group. Then every ergodic ${K}$-extension ${X}$ of ${Y}$ is equivalent to an ${H}$-uniform extension of ${Y}$ for some closed subgroup ${H}$ of ${K}$.

This theorem is usually stated for amenable groups ${G}$, but by using Theorem 1 (or more precisely, Corollary 3) the result is in fact also valid for arbitrary groups; we give the proof below the fold. (In the usual formulations of the theorem, ${X}$ and ${Y}$ are also required to be Lebesgue spaces, or at least standard Borel, but again with our abstract approach here, such hypotheses will be unnecessary.) Among other things, this theorem plays an important role in the Furstenberg-Zimmer structural theory of measure-preserving systems (as well as subsequent refinements of this theory by Host and Kra); see this previous blog post for some relevant discussion. One can obtain similar descriptions of non-ergodic extensions via the ergodic decomposition, but the result becomes more complicated to state, and we will not do so here.

A dynamical system is a space X, together with an action $(g,x) \mapsto gx$ of some group $G = (G,\cdot)$.  [In practice, one often places topological or measure-theoretic structure on X or G, but this will not be relevant for the current discussion.  In most applications, G is an abelian (additive) group such as the integers ${\Bbb Z}$ or the reals ${\Bbb R}$, but I prefer to use multiplicative notation here.]  A useful notion in the subject is that of an (abelian) cocycle; this is a function $\rho: G \times X \to U$ taking values in an abelian group $U = (U,+)$ that obeys the cocycle equation

$\rho(gh, x) = \rho(h,x) + \rho(g,hx)$ (1)

for all $g,h \in G$ and $x \in X$.  [Again, if one is placing topological or measure-theoretic structure on the system, one would want $\rho$ to be continuous or measurable, but we will ignore these issues.] The significance of cocycles in the subject is that they allow one to construct (abelian) extensions or skew products $X \times_\rho U$ of the original dynamical system X, defined as the Cartesian product $\{ (x,u): x \in X, u \in U \}$ with the group action $g(x,u) := (gx,u + \rho(g,x))$.  (The cocycle equation (1) is needed to ensure that one indeed has a group action, and in particular that $(gh)(x,u) = g(h(x,u))$.)  This turns out to be a useful means to build complex dynamical systems out of simpler ones.  (For instance, one can build nilsystems by starting with a point and taking a finite number of abelian extensions of that point by a certain type of cocycle.)

A special type of cocycle is a coboundary; this is a cocycle $\rho: G \times X \to U$ that takes the form $\rho(g,x) := F(gx) - F(x)$ for some function $F: X \to U$.  (Note that the cocycle equation (1) is automaticaly satisfied if $\rho$ is of this form.)  An extension $X \times_\rho U$ of a dynamical system by a coboundary $\rho(g,x) := F(gx) - F(x)$ can be conjugated to the trivial extension $X \times_0 U$ by the change of variables $(x,u) \mapsto (x,u-F(x))$.

While every coboundary is a cocycle, the converse is not always true.  (For instance, if X is a point, the only coboundary is the zero function, whereas a cocycle is essentially the same thing as a homomorphism from G to U, so in many cases there will be more cocycles than coboundaries.  For a contrasting example, if X and G are finite (for simplicity) and G acts freely on X, it is not difficult to see that every cocycle is a coboundary.)  One can measure the extent to which this converse fails by introducing the first cohomology group $H^1(G,X,U) := Z^1(G,X,U) / B^1(G,X,U)$, where $Z^1(G,X,U)$ is the space of cocycles $\rho: G \times X \to U$ and $B^1(G,X,U)$ is the space of coboundaries (note that both spaces are abelian groups).  In my forthcoming paper with Vitaly Bergelson and Tamar Ziegler on the ergodic inverse Gowers conjecture (which should be available shortly), we make substantial use of some basic facts about this cohomology group (in the category of measure-preserving systems) that were established in a paper of Host and Kra.

The above terminology of cocycles, coboundaries, and cohomology groups of course comes from the theory of cohomology in algebraic topology.  Comparing the formal definitions of cohomology groups in that theory with the ones given above, there is certainly quite a bit of similarity, but in the dynamical systems literature the precise connection does not seem to be heavily emphasised.   The purpose of this post is to record the precise fashion in which dynamical systems cohomology is a special case of cochain complex cohomology from algebraic topology, and more specifically is analogous to singular cohomology (and can also be viewed as the group cohomology of the space of scalar-valued functions on X, when viewed as a G-module); this is not particularly difficult, but I found it an instructive exercise (especially given that my algebraic topology is extremely rusty), though perhaps this post is more for my own benefit that for anyone else.